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A320658 Number of factorizations of A181821(n) into semiprimes. Number of multiset partitions, of a multiset whose multiplicities are the prime indices of n, into pairs. 5
1, 0, 1, 1, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 0, 3, 0, 0, 1, 0, 2, 1, 0, 0, 2, 0, 5, 2, 1, 3, 0, 0, 0, 1, 0, 6, 1, 0, 2, 4, 0, 0, 1, 0, 0, 1, 0, 9, 3, 0, 0, 2, 1, 0, 2, 0, 2, 0, 0, 0, 1, 1, 6, 15, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 6, 2, 0, 0, 1, 0, 17, 1, 0, 7, 2, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

The a(84) = 7 factorizations into semiprimes:

  84 = (4*4*9*35)

  84 = (4*4*15*21)

  84 = (4*6*6*35)

  84 = (4*6*10*21)

  84 = (4*6*14*15)

  84 = (4*9*10*14)

  84 = (6*6*10*14)

The a(84) = 7 multiset partitions into pairs:

  {{1,1},{1,1},{2,2},{3,4}}

  {{1,1},{1,1},{2,3},{2,4}}

  {{1,1},{1,2},{1,2},{3,4}}

  {{1,1},{1,2},{1,3},{2,4}}

  {{1,1},{1,2},{1,4},{2,3}}

  {{1,1},{2,2},{1,3},{1,4}}

  {{1,2},{1,2},{1,3},{1,4}}

MATHEMATICA

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];

bepfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[bepfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], PrimeOmega[#]==2&]}]];

Table[Length[bepfacs[Times@@Prime/@nrmptn[n]]], {n, 100}]

CROSSREFS

Cf. A001221, A001222, A007716, A007717, A056239, A112798, A181821, A305936, A318284, A320655, A320656, A320659.

Sequence in context: A083650 A138514 A286137 * A284966 A143540 A291336

Adjacent sequences:  A320655 A320656 A320657 * A320659 A320660 A320661

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 18 2018

STATUS

approved

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Last modified August 5 16:17 EDT 2021. Contains 346480 sequences. (Running on oeis4.)